Built on

O. Steinmann, Über den Zusammenhang zwischen den Wightmanfunktionen und der retardierten Kommutatoren
1960
Earlier work this paper cites.
O. Steinmann, Wightman-Funktionen und retardierten Kommutatoren. II
1960
Earlier work this paper cites.
Cambridge University Press, 1966
R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne, The Analytic S-Matrix · 1966
Earlier work this paper cites.
K. E. Cahill and H. P. Stapp, Optical Theorems and Steinmann Relations
1975
Earlier work this paper cites.
L. Brink, J. H. Schwarz, and J. Scherk, Supersymmetric Yang-Mills Theories
1977
Earlier work this paper cites.
F. Gliozzi, J. Scherk, and D. I. Olive, Supersymmetry, Supergravity Theories and the Dual Spinor Model
1977
Earlier work this paper cites.
M. T. Grisaru, H. N. Pendleton, and P. van Nieuwenhuizen, Supergravity and the S Matrix
1977
Earlier work this paper cites.
M. T. Grisaru and H. N. Pendleton, Some Properties of Scattering Amplitudes in Supersymmetric Theories
1977
Earlier work this paper cites.
S. J. Parke and T. R. Taylor, An Amplitude for n n Gluon Scattering
1986
Earlier work this paper cites.
F. A. Berends and W. T. Giele, Recursive Calculations for Processes with n Gluons
1988
Earlier work this paper cites.
V. P. Nair, A Current Algebra for Some Gauge Theory Amplitudes
1988
Earlier work this paper cites.
L. Magnea and G. F. Sterman, Analytic continuation of the Sudakov form-factor in QCD
1990
Earlier work this paper cites.
S. Catani, The Singular behavior of QCD amplitudes at two loop order
1998
Earlier work this paper cites.
M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry
2003
Earlier work this paper cites.
G. F. Sterman and M. E. Tejeda-Yeomans, Multiloop amplitudes and resummation
2003
Earlier work this paper cites.
Z. Bern, L. J. Dixon, and D. A. Kosower, On-shell recurrence relations for one-loop QCD amplitudes
2005
Earlier work this paper cites.
Z. Bern, L. J. Dixon, and V. A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond
2005
Earlier work this paper cites.
A. B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry
2005
Earlier work this paper cites.
W. Stein and D. Joyner, SAGE: System for algebra and geometry experimentation
2005
Earlier work this paper cites.
Z. Chen and A. Storjohann, A blas based c library for exact linear algebra on integer matrices
2005
Earlier work this paper cites.
Z. Bern, L. J. Dixon, and D. A. Kosower, Bootstrapping multi-parton loop amplitudes in QCD
2006
Earlier work this paper cites.
C. F. Berger, Z. Bern, L. J. Dixon, D. Forde, and D. A. Kosower, Bootstrapping One-Loop QCD Amplitudes with General Helicities
2006
Earlier work this paper cites.
J. Drummond, J. Henn, V. Smirnov, and E. Sokatchev, Magic identities for conformal four-point integrals
2007
Earlier work this paper cites.
Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower, and V. A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
2007
Earlier work this paper cites.
Z. Bern, J. Carrasco, H. Johansson, and D. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops
Original
2007
Earlier work this paper cites.
L. F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT
Original
2007
Earlier work this paper cites.
L. F. Alday and J. M. Maldacena, Gluon scattering amplitudes at strong coupling
Original
2007
Earlier work this paper cites.
N. Arkani-Hamed and J. Trnka, The Amplituhedron
Original
2007
Earlier work this paper cites.
J. Drummond, G. Korchemsky, and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops
Original
2008
Earlier work this paper cites.
A. Brandhuber, P. Heslop, and G. Travaglini, MHV amplitudes in 𝒩 = 4 \mathcal{N}=4 super Yang-Mills and Wilson loops
Original
2008
Earlier work this paper cites.
J. Drummond, J. Henn, G. Korchemsky, and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality
Original
2008
Earlier work this paper cites.
Z. Bern, L. Dixon, D. Kosower, R. Roiban, M. Spradlin, et al., The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
Original
2008
Earlier work this paper cites.
J. Drummond, J. Henn, G. Korchemsky, and E. Sokatchev, The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude
Original
2008
Earlier work this paper cites.
J. Drummond, J. Henn, G. Korchemsky, and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude
Original
2009
Earlier work this paper cites.
V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm
2009
Earlier work this paper cites.

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Original
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Later among the works it cites.
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A. B. Goncharov, A simple construction of Grassmannian polylogarithms
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2013
Later among the works it cites.
L. J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops
Original
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Later among the works it cites.
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2016

Heptagons from the Steinmann Cluster Bootstrap

Dixon, Lance J., Drummond, James, Harrington, Thomas et al.

Understand

We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish.

These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planar $\mathcal{N} = 4$ supersymmetric Yang-Mills theory, making higher-loop contributions to these amplitudes more computationally accessible.
In particular, dual superconformal symmetry and well-defined collinear limits suffice to determine uniquely the symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes.
We also show that at three loops, relaxing the dual superconformal ($\bar{Q}$) relations and imposing dihedral symmetry (and for NMHV the absence of spurious poles) leaves only a single ambiguity in the heptagon amplitudes.

Built on

O. Steinmann, Über den Zusammenhang zwischen den Wightmanfunktionen und der retardierten Kommutatoren
1960
Earlier work this paper cites.
O. Steinmann, Wightman-Funktionen und retardierten Kommutatoren. II
1960
Earlier work this paper cites.
Cambridge University Press, 1966
R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polkinghorne, The Analytic S-Matrix · 1966
Earlier work this paper cites.
K. E. Cahill and H. P. Stapp, Optical Theorems and Steinmann Relations
1975
Earlier work this paper cites.
L. Brink, J. H. Schwarz, and J. Scherk, Supersymmetric Yang-Mills Theories
1977
Earlier work this paper cites.
F. Gliozzi, J. Scherk, and D. I. Olive, Supersymmetry, Supergravity Theories and the Dual Spinor Model
1977
Earlier work this paper cites.
M. T. Grisaru, H. N. Pendleton, and P. van Nieuwenhuizen, Supergravity and the S Matrix
1977
Earlier work this paper cites.
M. T. Grisaru and H. N. Pendleton, Some Properties of Scattering Amplitudes in Supersymmetric Theories
1977
Earlier work this paper cites.
S. J. Parke and T. R. Taylor, An Amplitude for n n Gluon Scattering
1986
Earlier work this paper cites.
F. A. Berends and W. T. Giele, Recursive Calculations for Processes with n Gluons
1988
Earlier work this paper cites.
V. P. Nair, A Current Algebra for Some Gauge Theory Amplitudes
1988
Earlier work this paper cites.
L. Magnea and G. F. Sterman, Analytic continuation of the Sudakov form-factor in QCD
1990
Earlier work this paper cites.
S. Catani, The Singular behavior of QCD amplitudes at two loop order
1998
Earlier work this paper cites.
M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry
2003
Earlier work this paper cites.
G. F. Sterman and M. E. Tejeda-Yeomans, Multiloop amplitudes and resummation
2003
Earlier work this paper cites.
Z. Bern, L. J. Dixon, and D. A. Kosower, On-shell recurrence relations for one-loop QCD amplitudes
2005
Earlier work this paper cites.
Z. Bern, L. J. Dixon, and V. A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond
2005
Earlier work this paper cites.
A. B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry
2005
Earlier work this paper cites.
W. Stein and D. Joyner, SAGE: System for algebra and geometry experimentation
2005
Earlier work this paper cites.
Z. Chen and A. Storjohann, A blas based c library for exact linear algebra on integer matrices
2005
Earlier work this paper cites.
Z. Bern, L. J. Dixon, and D. A. Kosower, Bootstrapping multi-parton loop amplitudes in QCD
2006
Earlier work this paper cites.
C. F. Berger, Z. Bern, L. J. Dixon, D. Forde, and D. A. Kosower, Bootstrapping One-Loop QCD Amplitudes with General Helicities
2006
Earlier work this paper cites.
J. Drummond, J. Henn, V. Smirnov, and E. Sokatchev, Magic identities for conformal four-point integrals
2007
Earlier work this paper cites.
Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower, and V. A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory
2007
Earlier work this paper cites.
Z. Bern, J. Carrasco, H. Johansson, and D. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops
Original
2007
Earlier work this paper cites.
L. F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT
Original
2007
Earlier work this paper cites.
L. F. Alday and J. M. Maldacena, Gluon scattering amplitudes at strong coupling
Original
2007
Earlier work this paper cites.
N. Arkani-Hamed and J. Trnka, The Amplituhedron
Original
2007
Earlier work this paper cites.
J. Drummond, G. Korchemsky, and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops
Original
2008
Earlier work this paper cites.
A. Brandhuber, P. Heslop, and G. Travaglini, MHV amplitudes in 𝒩 = 4 \mathcal{N}=4 super Yang-Mills and Wilson loops
Original
2008
Earlier work this paper cites.
J. Drummond, J. Henn, G. Korchemsky, and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality
Original
2008
Earlier work this paper cites.
Z. Bern, L. Dixon, D. Kosower, R. Roiban, M. Spradlin, et al., The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory
Original
2008
Earlier work this paper cites.
J. Drummond, J. Henn, G. Korchemsky, and E. Sokatchev, The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude
Original
2008
Earlier work this paper cites.
J. Drummond, J. Henn, G. Korchemsky, and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude
Original
2009
Earlier work this paper cites.
V. V. Fock and A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm
2009
Earlier work this paper cites.